Effectiveness of Rotation-free Triangular and Quadrilateral Shell Elements in Sheet-metal Forming Simulations M. Brunet and F. Sabourin LaMCoS, Institut National des Sciences Appliquées de Lyon, 20 Avenue Einstein, 69621 Villeurbanne, France,
[email protected] Abstract. This paper is concerned with the effectiveness of triangular 3-node shell element without rotational d.o.f. and the extension to a new 4-node quadrilateral shell element called S4 with only 3 translational degrees of freedom per node and one-point integration. The curvatures are computed resorting to the surrounding elements. Extension from rotationfree triangular element to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. Two numerical examples with regular and irregular meshes are performed to show the convergence and accuracy. Deepdrawing of a box, spring-back analysis of a U-shape strip sheet and the crash simulation of a beam-box complete the demonstration of the bending capabilities of the proposed rotation-free triangular and quadrilateral elements.
propose an attractive shell element, the one-point quadrature is achieved at the center for the membrane strains which are superposed to the already superposed bending strains in the centered co-rotational local frame. The membrane hourglass control is obtained by the perturbation or physical stabilization procedure.
INTRODUCTION Bending elements, plates and shells, using only out-of-plane translational degrees of freedom are not new but, so far as we know for shell analysis, with triangular shape only, the state of the art has been given by Oñate and Zarate in [1]. In the context of sheet-metal forming analysis, Brunet and Sabourin [2] [3] have proposed an approach to compute the constant curvature field within each triangle in terms of the six out-of-plane displacements of the neighbouring nodes. More recently, Sabourin and Fayet [4] have shown the kinematical aspects of the patch formed by the four triangular elements using the reciprocal screws formulations. In this paper, the formulation is based on the area quadrilateral coordinates interpolation of the rigid body angles in term of the out-of-plane nodal deflections of the patch surrounding the element and related to the bending angles along the four sides of the quadrilateral. Since warping modes are possible modes of deformation, two additional bending angles must be considered along the diagonals of the quadrilateral. After frame transformations in the co-rotational local frame with a superposition of the strains defined separately in normal directions to each side and diagonal, the [B] (3x36) bending gradient matrix is obtained for the quadrilateral in its surrounding patch. In order to
ROTATION-FREE TRIANGLE “S3” Let us consider the triangular element (1,2,3) of Fig. 1 and its immediate neighbours and, for example, triangle (3,2,5) sharing the boundary 2-3 with the element (1,2,3). After the rotational angle α1 for triangle (1,2,3) and α 5 for triangle (3,2,5), the two lines perpendicular to their respective mid-plane give the point C. The sketching of CH defines the bending angles θ1 and θ5 whose sum is assumed to be equal to
α1 + α 5 such that: 2θ α1 + α5 = θ1 + θ5 ; ε n1 = −z 2θ1 = −z 5 h1 h5
(1)
{ε n } = z[H]{α}
(2)
CP778 Volume A, Numisheet 2005, edited by L. M. Smith, F. Pourboghrat, J.-W. Yoon, and T. B. Stoughton © 2005 American Institute of Physics 0-7354-0265-5/05/$22.50
745
w6 w1 w4
ROTATION-FREE QUADRANGLE “S4”
6
4
w3
1 3
w2
4
w5
t3
A3
5 C
n4
x
w12
θ1
α5
5
H 3
α5
α1
h5
9 w3
z 4
w8
y
3
x
w7
w2
1
2
8
7
5
FIGURE 2. Projected in-plane areas and quadrilateral patch.
If it is noted that ε n , ε n , ε n , ε n are the normal strains at any depth z at the mid-point of the sides joining (23),(3-4),(4-1) and (1-2) respectively, it is assumed for example that: 1
FIGURE 1. Triangular patch, bending and rigid body angles on common side 2-3.
{α} = [C]{w}
w4
w1 w6
w9
10
6
2
Assuming that
w10
11
12 w5
α1
h1
2
w11
α1 + α 5 θ5
1
t4
t1 n1
A1
A4
y
3
A2
P
1 2
n2
t2
n3
relates the rigid body
rotations to the normal displacements w i , this purely kinematical relation can be obtained by several ways. The interpolation by the triangular area coordinates A i /A are used here since the extension to the 4-node quadrilateral with quadrilateral area functions will be straightforward:
ε1n = −2z where
2
3
4
α 2,3, 4,1 + α3, 2, 7 ,8 h 4,1 + h1,1 + h 7 ,1 + h 8,1
α i , j, k , m means the rotational angle α at mid-
side (i-j) for the quadrilateral (i,j,k,m).
A A A w = 1 w1 + 2 w 2 + 3 w 3 A A A
(3)
w i = CosXz Ui + CosYz Vi + Cos Zz Wi
(4)
(5)
which leads to the bending strains in the local frame:
{δε} = z[R ][H][C]{δw} = z[B ]{δw} b
h i , j is the
corresponding height from node i to side (j) where (1)=(2-3), (2)=(3-4), (3)=(4-1) and (4)=(1-2). Since quadrilateral may undergoes warping kinematic modes of deformation, additional bending strains must be taken account in the quadrilateral to avoid singular stiffness matrix. A possible way to achieve this requirement is to consider the normal bending strains along the two diagonals named (5)=(2-4) and (6)=(3-1) such that for example:
Then the rigid body rotations:
H H α i = grad ( w).(−ni )
(7)
ε5n = z
(6)
α 2, 4,1 + α 4, 2,3 h1 + h 3
{{ ε n } = z[{ H ]{{ α}
where [R] is the transformation matrix between the director cosines of the outward pointing normals n i and the local x,y in plane coordinates.
6 x1
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6 x12 12 x1
(8)
(9)
α i , j, k is the rotational angle α along diagonal
where
B F
(i-j) for the internal triangle (i,j,k), h k being the height from node k to diagonal (i-j). Assuming that:
A
4
w = å N k w k where the shape functions in terms of
C
k =1
the area coordinates by:
Ni =
D
L i of the quadrilateral are given
1 é Li L j L k + L m − ê1 + + 4 êë g j g m gi
(10)
ù g + k (g i Li L j − g jL jL k + g k L k L m − g m L m Li )ú g0 û FIGURE 3. Uniform torsion with regular and irregular 2x2 and 8x8 meshes for example.
Applying as for the triangular element: H H α i = grad ( w).(−ni ) provides the required [C] matrix of {α} = [C]{w} and then the local [B] bending matrix
Pinched Hemisphere With 18° Hole
[{] [{][{][{]
as: Bb = R H C . The one point quadrature is 3 x12
A hemispherical shell, with an 18° hole at the top as shown on Fig. 4, is under the action of two inward and outward forces 90° apart (F=1). Young’s modulus E = 6 .825 .10 7 , Poisson’s ratio ν = 0.3 , radius R=10 and the thickness a=0.04 are used.
3 x 6 6 x1212 x12
achieved at the centred local frame for the membrane strains using the perturbation or the physical stabilisation and a warping correction is performed. The integrated through the thickness (12x12) local matrix becomes a (36x36) symmetric bending stiffness matrix for the full patch in the global system of coordinates using the projection Eq. (4).
NUMERICAL EXAMPLES Bending Patch Test A uniform torsion is considered where a square plate (100x100) of thickness 1 is simply supported at three of its corners as shown in Fig. 3 and a vertical load F=1 is applied at the last one. In table (1) is shown the convergence of the quadrilateral “S4” element. TABLE 1. Corner Displacement Bending Patch Test. Mesh Number Load Deflection Normalized Analytical 0.39000 1.00000 Regular Meshes 0.39000 1.00000 Irregular 2x2 0.41535 1.06500 Irregular 4x4 0.38785 0.99448 Irregular 8x8 0.38879 0.99689 Irregular 16x16 0.38960 0.99897 FIGURE 4. Initial and deformed meshes for the non linear calculations F=100.
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For the non-linear case, the loads applied are increased by two orders of magnitude (F=100) in 20 steps using the updated Lagrangian formulation with Newton’s equilibrium iterations.
shear strain and five degrees of freedom per node and named here Q4γ24. The great mesh size sensitivity is mainly due to the rotational d.o.f. because they are not controlled by the contact algorithms.
TABLE 2. Pinched Hemisphere Results For F=1 Linear Load Displacement One quadrant 4x4mesh 8x8mesh 16x16mesh Reference [5] 0.092530 0.092697 0.092990 “S4” membrane 0.105102 0.095036 0.092167 1x1 integration “S4” membrane 0.097532 0.084574 0.073348 2x2 integration Triangle “S3” 0.085258 0.087506 0.08925
TABLE 3. Pinched Hemisphere Results For F=100 Non-Linear Inward-Outward Deflections One quadrant 4x4mesh 8x8mesh 16x16mesh “S4” membrane In: 6.120 In: 5.894 In: 5.826 1x1 integration Out: 3.324 Out: 3.117 Out: 3.066 “S4” membrane In: 3.310 In: 4.651 In: 5.331 2x2 integration Out: 1.178 Out: 2.861 Out: 2.918 Triangle “S3” In: 6.006 In: 5.576 In: 5.736 Out:3.441 Out: 3.364 Out: 3.335
FIGURE 6. Mesh size sensitivity with conventional assumed stain element Q4γ24
From tables 2 and 3, it can be seen that the quadrilateral “S4” element is vulnerable to membrane locking when the membrane behaviour is fully (2x2) integrated. One-point (1x1) integration with hourglass stabilisation of the membrane strains gives much better results. The results given by the rotation-free triangular “S3” and quadrilateral “S4” elements are in good agreement with many similar results reported in the literature.
FIGURE 7. Mesh size sensitivity with rotation-free “S3” TABLE 4. Residual Wall Radius (mm) Versus Mesh Size Rotation-free Triangle “S3” Mesh: 1.5mm 2.5mm 3.5mm 5.0mm 5 “Lobatto” integration points Radius: 82.4 82.5 89.7 91.2 Global plasticity Radius: 85.8 86.3 93.6 89.9 Experimental NUMISHEET’93 Mini: 75.7 Average: 88.7 Maxi: 99.0
Springback Analysis
Fig. (7) and table (4) show the results obtained with the rotation-free triangle “S3” for the case of the aluminum alloy, see [2],[6],[7] for details. The global plasticity is performed with the yield surface f given in term of the resulting efforts and moments as [6], [8]:
f = Q n + Q m + 32 Q 2mn − 1 = 0 where: (11) Q N = N 2x + N 2y − N x .N y + 3.N 2xy FIGURE 5. U-bending process before and after springback
Q M = M 2x + M 2y − M x .M y + 3.M 2xy
This U-bending test was used as a benchmark problem for NUMISHEET’93 [2] to compare various FEM codes. On Fig. (6), the example shows the mesh size sensitivity of conventional shell elements such as the linear 4-node shell element with assumed transverse
Q MN = M x .N x + M y .N y − 12 M x .N y − 12 M y .N x + 3.M xy .N xy 4 Q MN 16 Q M 1 QN Q mn = Qm = Qn = h 3 σ s2 h 4 σ s2 h 2 σ s2
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and n=0.2637. Due to symmetry, only one quarter of the geometry is modeled by a regular 25x25 mesh. The simulation is implicit using the updated Lagrangian formulation and Newton equilibrium iterations with an elastic-plastic consistent tangent modulus (algorithmic tangent). A geometric constraint combined with friction boundary condition is imposed for the contact treatment and is enforced by a penalty method. In the penalty method, the impenetrability condition is imposed as a penalty normal traction along the contact surface via a penalty parameter β, the impenetrability condition is satisfied only approximately. The stiffness matrix for contact and the corresponding contact forces are directly proportional to the penalty parameter. Table 5 lists the amounts of draw-in in the X , Y and diagonal directions named respectively by DX ,DY and DD at punch travels of 15mm and 40 mm.
Deep Drawing of a Square Box The present example concerns a square cup drawing with a initial square blank proposed as a basic benchmark for NUMISHEET’93, see [2]. The punch is a 70x70 mm square and the die a 74x74 mm square, the clearance is 2 mm. The punch corner radius is 10 mm with a shoulder radius of 8 mm while the die corner radius is 12 mm with a shoulder radius of 5 mm.
TABLE 5. Draw-in for Mild-Steel Square Cup (DX, DY, DD unit: mm) 15 mm Punch Travel 40 mm Punch Travel DX DY DD DX DY DD 6.11 6.09 2.89 26.77 26.43 15.01 “S4” 6.34 6.36 3.42 28.93 29.61 17.07 “S3” 6.17 6.12 3.24 27.96 27.95 15.86 Exp. (Exp.: Average Experimental NUMISHEET’93)
FIGURE 8. Rigid tools of the deep-drawing process
Crash of a Box Beam The rotation-free element “S3” has been implemented in the commercial explicit code “RADIOSS” of MECALOG (France) under the name “S3N6” and has been compared to two other 3-node triangles: “DKT18” for Discrete Kirchhoff Triangle and “T3C0” for the Belytschko-Tsay triangular element. A box beam of rectangular cross section of (50.8x38.1mm), thickness 0.914mm, length=209mm, is impacted by an infinite mass with a constant velocity of 1.27 m/s. The steel behavior is: E=2.1e5 0.5 Mpa, ν=0.3 and hardening σ = 206 + 450 ε Mpa, density: ρ=7.8e-3. The top end of the box beam was attached to the impacting rigid-wall with a rigid link having the same translation and rotation velocities according to the experimental conditions. A random noise of 1.0 e-3 has been given to the mesh.
FIGURE 9. Deformed “S4” mesh for 40 mm punch travel
The square flat shape of the blank is 150x150 mm with 0.78 mm thickness for the material mild-steel. During the forming process, the die is fixed and the blankholding force is 19.6 KN with a Coulomb friction coefficient of 0.144. Values for characterizing the mild-steel are E=206000 Mpa, ν = 0.3 and the Lankford’s coefficients are R 0 = 1.79 , R 90 = 2.27 ,
For the mesh size of 3x3mm “S3N6” and “DKT18” have almost the same deformation, i.e.: three plies in one side and two folds in another. “T3C0” presents one more fold in the second side and seems to be a little too stiff according to the force-displacement curve, see Fig. (10) and (11). “DKT18” and “S3N6” have also almost the same stiffness response and near to the experiment result according to the force-displacement curve.
R 45 = 1.51 required for the Hill’48 anisotropic yield
criterion in plane-stress state. In the reference rolling direction, the form of the isotropic hardening law is:
σ = B(c + ε p ) n with B=567.29 Mpa, c=0.007127
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DKT18
T3C0
S3N6
FIGURE 13. Normal force in the impacting rigid-wall versus displacement (5x5 mm mesh size)
FIGURE 10. 3x3 mm mesh size. Deformed beam at t=0.08s for an impacting rigid-wall displacement of 100mm.
For the 5x5mm mesh size, the three elements seem to be stiffer according to the force-displacement curve especially for “T3C0”, see Fig. (13). They are more sensitive to the mesh sizes, and the buckle modes change with coarse mesh especially for “S3N6”in this case.
REFERENCES 1. Oñate E, Zarate F. Rotation-free triangular plate and shell elements. International Journal for Numerical Methods in Engineering 47, 557-603 (2000). 2. Brunet M, Sabourin F. “A simplified triangular shell element with necking criterion for 3D. sheet forming analysis” in NUMISHEET’93 edited by Makinouchi A, Nakamachi E, Oñate E, Wagoner RH, Riken, Tokyo 1993; pp. 229-238. 3. Sabourin F, Brunet M. Analysis of plates and shells with a simplified three-node triangular element Thin Walled Structures 21, 209-223 (1995). 4. Sabourin F, Fayet M. Incorporation of theory of screws in a new finite element Journal of Multibody Dynamic 215, 61-73 (2001). 5. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model Part 3. Computer Methods in Applied Mechanics and Engineering 79, 91126 (1990). 6. Sabourin F., Brunet M., Vives M. “3D springback analysis with a simplified 3-node triangular element” in NUMISHEET’99 edited by Gelin J.C. and Picart P., University of Franche-Comté, France 13-17 September 1999, pp. 17-22. 7. Sabourin F, Morestin F, Brunet M. “Effect on non-linear kinematic hardening on spring-back analysis” in NUMISHEET’02 edited by Yang DY., Jejun Island, South-Korea 21-25 October 2002, pp. 79-84. 8. Sabourin F, Vives M. Eléments finis triangulaires pour la simulation de l’emboutissage Revue Européenne des éléments finis 10, 7-53 (2001).
FIGURE 11. Normal force in the impacting rigid-wall versus displacement (3x3 mm mesh size)
DKT18
T3C0
S3N6
FIGURE 12. 5x5 mm mesh size. Deformed beam at t=0.08s for an impacting rigid-wall displacement of 100mm
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